Ideals Of Quantum Superalgebras U_q(osp(1,2))

As a newly-arising branch of mathematics in the1980s, Quantum group has been greatly developed in recent decades. Due to its profound physics background and signif-icance, Quantum group has a close relationship with quantum mechanics, which endows it with abaundant theories and a wide range of applications. So far, ideal classification of the vast majority of the Enveloping Algebra is still not been resolved but the com-plete lists of some enveloping algebra and their quantified enveloping algebra, such as U(sl2), Uq(sl2)) and Uq(sl2), have been abtained.In this thesis, we investigate the ideals generated by elements which are locally finite. It turns out that these ideals can be generated by four highest weight vectors under the adjoint action, comparbale to the non-super quantized enveloping algebra. Further, these ideals can be generated by two elements(one is a highest vector) which are unique under the expession given in the thesis. This is theoretically significant for the research of prime ideals and primitive ideals.Our main results can be stated as follows:(1) Let u be a highest weight vector of U under the adjoint action. Then there exists f(c)∈k[c] and n∈No=N-{0} s.t. u=EnSδnf(c). Moreover,[EnSδnf(c)]≌V(2n).(2) Let I be an ideal of U which is generated by elements which are locally fi-nite. Then there exist n,m∈N and2|n,2|m and f(c), g(c), f’(c), g’(c) sat-isfying g(c)|fn0(c), g’(c)|fm0(c) and φn(c)|g(c) or φn-1(c)|g-(c) and φm(c)|g’(c) or φm-1(c)|g’(c) such that I=(Enf(c), f(c)g(c))+(Emf’(c), f’(c)g’(c))S.(3) Let I be an ideal of U which is generated by elements which are locally finite. Then there exist odd number r∈No, δ=0or1and h(c), g(c)∈k[c] satisfying g(c)|fro(c) and φr(c)|g(c) or φr-1(c)|g(c), such that I=h(c)(ErS, g(c)Sδ), Besides, r and h(c), g(c) satisfying the required conditions are unique.